CAT 2025 Slot 2 QA Question 3: The equations $3 x^2 - 5 x + p = 0$ and $2 x^2 - 2 x + q = 0$ have one common root. The sum of the other roots of these two equations is

Question

The equations $3 x^2 - 5 x + p = 0$ and $2 x^2 - 2 x + q = 0$ have one common root. The sum of the other roots of these two equations is

Accepted Answer

Let the common root be $r$. Let the other roots of the first and second equations be $u$ and $v$, respectively.

From the sum of roots,
$r + u = \dfrac{5}{3}$
$r + v = 1$
Adding the other roots, $u + v = \dfrac{8}{3} - 2r$

Since $r$ is a root of both equations,
$3 r^2 - 5 r + p = 0$
$2 r^2 - 2 r +

The equations 3x2−5x+p=03 x^2 - 5 x + p = 03x2−5x+p=0 and 2x2−2x+q=02 x^2 - 2 x + q = 02x2−2x+q=0 have one common root. The sum of the other roots of these two equations is